How do I animate still images in After Effects to create School of Life-style animation?

It's all about preparing the artwork for the project and learning how to set keyframes. You can automate some of the moves using expressions, but most of the work is in the planning and prep.

To make the animations really stand out you should be very familiar with the Illusion of Life principals that some Disney animators put into a book a long time ago. Applying those principals well takes a lot of practice.

You'll also want to break up your project into phrases or sentences. Putting together an animation like the one you pointed to in a single comp will be almost impossible to work on in just a couple of hours. Each comp should be just a few seconds long covering just a single phrase or sentence, then you should render them and do the final edit in Premiere Pro so you can fine-tune the audio and video accurately.

If you're using photo pieces like this, you'll be taking one of two approaches (or possibly combining them):
1. Cut your photo into pieces so you can separately move arms, etc.
2. Use the puppet tool to manipulate different portions of the images.

There will be a lot of prep work in Photoshop - think about how you'll need to dissect an image to get the movements you'll need in AE.

For example, if you want a person to raise their arm, you'll probably want to split them into two layers in Ps - a body, and then the arm isolated. Bring that into AE, set the arm's anchor point to the shoulder and parent the arm to the body. Use rotation to move the arm. You're essentially making a paper doll, if that makes sense.

For something like your bat, puppet pins are probably a better answer. Look up tutorials on the Puppet Tool, but in short - make some pins on the points you want to control, then animate them accordingly.

There are definitely a lot of tutorials available on both of these - hopefully the above will give you some better search terms for what you're after. Let us know if you need more guidance!